The trace theorem

The trace theorem

W

(Ω

) ∋ f 7→ ∇

f ∈ W

p

( ∂ Ω

)

revisited

Peter Weidemaier

Abstract. Filling a possible gap in the literature, we give a complete and readable proof of this trace theorem, which also shows that the imbedding constant is uniformly bounded forT↓0. The proof is based on a version of Hardy’s inequality (cp. Appendix).

Keywords: trace theory, anisotropic Sobolev spaces Classification: 46E35

Introduction.

The imbedding theorem described in the title can be found in LADYSHENS- KAYA et al. [L/S/U, Chapter II, Lemma 3.4]. However, none of the references cited there seems to contain a complete proof. The theorem is also stated in IL’IN [I, Theorem 8.4]; but there too, no proof is given. Things look even worse, if we ask for the dependence of the imbedding constantc(T) on the height T of the space- time cylinder (for smallT). In some applications of this trace theorem to nonlinear problems, one needsc(T)≤c0 for allT small (cf. WEIDEMAIER [W], particularly the Appendix). However, the formulation in IL’IN [I, Theorem 8.4], exhibits an explosion of c(T) forT ↓0. To settle things, we shall give in this note a detailed proof for the imbedding, which also shows the uniformity ofc(T) forT ↓0.

The paper is organized as follows: in Chapter 1 we deduce an integral representa- tion for∇xf in terms of∂tf, ∂_x²f, which is the basis for the estimates in Chapter 2.

Let us fix the notation: Ω_T := Ω×(0, T) with the typical point (x, t)∈Ω_T; here Ω⊂IRⁿ. The prime characterizes (n−1)-dimensional quantities : thus we writex∈ IRⁿasx= (x^′, x_n), x^′∈IRⁿ⁻¹;Qⁿ⁻¹(a^′, b^′) is the open parallelepipedQ_n−1

j=1(a_i, b_i), when a^′ = (a₁, . . . , a_n−1), b^′ = (b₁, . . . , b_n−1);Qⁿ⁻¹(λ) := Qⁿ⁻¹(−λ1^′, λ1^′) for λ∈IR; here

1^′ := (1,· · ·,1) ∈ INⁿ⁻¹;Q₊ⁿ(λ) := Qⁿ⁻¹(λ)×( 0, λ) ; the superscript ˇ always indicates the deletion of a coordinate (the n-th. one, if not further specified) , e.g.

ˇi

y= (y₁,· · ·, y_i−1, y_i+1,· · · , yn) (1≤i≤n) and ˇQⁿ⁺¹(a, b) :=Q_n+1

i6=ni=1

(a_i, b_i) . W_p^2,1(Ω_T) := {u|∂_x^αu, ∂_tu (distr. sense) ∈ L_p(Ω_T) ∀ |α| ≤ 2} with the obvious norm.

I thank Prof. V.A. Solonnikov, Leningrad, for valuable hints

For a bounded domain Ω ⊂ IRⁿ, ∂Ω ∈ C² means that ∂Ω is a C²-hypersurface.

The spaces W_p^α,β(∂Ω_T) (α, β ∈ (0,1)) are defined as usual, via a partition of unity on ∂Ω, and using local charts. We use the notation c^∗ to emphasize the non-dependence of the constantc on the quantityT (forT small).

1. Integral representation.

Our starting point is an integral representation for ∂^νf in terms of f: if f is smooth and defined on Qⁿ⁻¹(−λ1^′,2λ1^′)×[ 0,2λ]×[ 0,3T], then we have (cf.

IL’IN/ SOLONNIKOV [I/S, p. 70, (6)] withm_i= 0, k_i =l_i)

∂^νf(x, t) = A T^r

Z . . .

Z

Qⁿ⁺¹(0, T^κ)

f((x, t) +y)Π(y, T)dy+

+

n+1

X

i=1

B_i Z _T

0

v^−(1+r) Z

. . . Z

Qⁿ⁺¹(0, v^κ)

f((x, t) +y)Π_i(y, v)∂ˇⁱ _i^liψ_i(y_i, v)dy dv

for (x, t)∈Q₊ⁿ(λ)×[ 0, T], T ≤T₀(λ) andν_j ≤l_j−1, where (cp. [I/S, pp. 69–70])

Π(y, T) :=

n+1

Y

j=1

∂_j^ljχ_j(y_j, T)

χ_j(y_j, T) :=y^l_j^j−νj−1 Z _T^κj

yj

(T^κj−s)^µjs^λjds ,

Π_i(y, v) :=ⁱˇ

n+1

Y

j=1 j6=i

∂_j^ljχ_j(y_j, v),

ψ_i(y_i, v) :=y_i^li+λi−νi·(v^κi−y_i)^µi

with certain parametersµ_j, λ_j ∈IN and certainA, B_i∈IR; hereT^κ:=

(T^κ1,· · · , T^κn+1), r:=κ·(1 +λ+µ), 1 := (1,· · · ,1)∈INⁿ⁺¹.

In the sequel we fixl:= (2,· · · ,2,1)∈INⁿ⁺¹, κ= (κ^′, κn, κ_n+1) := ¹_l = (¹₂,· · · ,¹₂,1) and choose the parametersµ_j, λ_jso large that∂_j^kψ_j(y_j, v) vanishes fory_j= 0, y_j= T^κj, 1≤k≤l_j. Hence, integrating by parts and introducing K_i(y, v) := Π_i(y, v)ˇⁱ ψ_i(y_i, v) (0≤y_i≤v^κi) , we have shown that

(1.1) ∂^νf(x, t) = A T^r

Z . . .

Z

Qⁿ⁺¹(0, T^κ)

f((x, t) +y)Π(y, T)dy+

+

n+1

X

i=1

B˜_i Z T

0

v^−(1+r) Z

. . . Z

Qⁿ⁺¹(0, v^κ)

∂_i^lif((x, t) +y)K_i(y, v)dy dv.

The kernels Π, K_iin this representation satisfy (uniformly w.r.t. y∈Qⁿ⁺¹(0, v^κ))

|∂_y^αΠ(y, v)| ≤c·vr−κ·(1+ν+α) ∀ |α| ≤2 (1.2)

|∂_n+1^s K_i(y, v)| ≤c·y_n^ε·vr+1−κ·(1+ν)−εκn−s

(1.3)

(∂n+1 :=∂yn+1,0≤s≤1, 1≤i≤n+ 1, ε∈[0,1)).

For the proof of these two inequalities, we first note that∂_j^lj+αjχ_j(y_j, v) is a linear combination of terms of the form (v^κj−y_j)^ρ1y^ρ_j²withρ₁+ρ₂=µ_j+λ_j−ν_j−α_j, ρ₂>

0 (forλ_j large) and consequently

|∂_j^lj+αjχ_j(y_j, v)| ≤c·y^ε_j·v^{−κj(ε+αj)}·v^{κj(µj+λj−νj)} (0≤y_j ≤v^κj) for arbitraryε∈[0,1[ ; this implies (for 1≤k≤n−1)

|∂_n+1^s Π_k(^ky, v)| ≤ˇ c·y^ε_n·v^{−κnε−κn+1·s}·v^{κ(µ+λ−ν)−κkδk}

|∂_n+1^s Π_n(ⁿy, v)| ≤ˇ c·v^−κn+1·s·vκ·(µ+λ−ν)−κnδn

|Π_n+1(ⁿ⁺¹y , v)| ≤ˇ c·y^ε_n·v^−κn·ε·vκ·(µ+λ−ν)−κn+1δn+1, whereδ_j:=µ_j+λ_j−ν_j. The definition ofψ_i easily implies

|ψ_k(y_k, v)| ≤c·v^{κk·(lk+δk)}

|ψ_n(y_n, v)| ≤c·y^ε_n·v^−κn·ε·v^{κn·(ln+δn)}

|∂_n+1^s ψ_n+1(y_n+1, v)| ≤c·v^−s·κn+1·v^{κn+1·(ln+1+δn+1)} ;

since K_i(y, v) = Π_i(y, v)ψˇⁱ _i(y_i, v), κ_il_i = 1 (1 ≤ i ≤ n+ 1), κ_n+1 = 1, r = κ·(1 +λ+µ), these formulas yield (1.3). For (1.2) compare IL’IN/ SOLONNIKOV [I/S, p. 72].

2. Estimates.

Our aim in this chapter is to prove the imbedding W_p^2,1(Ω_T) ∋ f 7→ ∇_xf ∈ W¹⁻

1

p,¹₂(1−¹_p)

p (∂Ω_T) with the imbedding constantc^∗independent ofT (forT small);

here we let Ω be a bounded domain in IRⁿwith boundary of the classC². Flattening the boundary locally, it is no restriction to assume that Ω is a cube i.e. Ω =Qⁿ₊(λ).

SinceC²(Q₊ⁿ(λ)×[0, T]) is dense inW_p^2,1(Qⁿ₊(λ)×(0, T)) (cf. R ´AKOSN´IK [R, The- orem 3]) and since the Hestenes-Whitney extension method (cf. ADAMS [A, p. 83]) yields a linear continuous extension operator

E_T : W_p^2,1(Qⁿ₊(λ)×(0, T))→W_p^2,1(Qⁿ₊(2λ)×(0,2T)) with E_T(C²(Q₊ⁿ(λ)×[0, T]))⊂C²(Qⁿ₊(2λ)×[0,2T]) and

kE_Tk_W2,1

p (Qⁿ₊(λ)×(0,T))→Wp^2,1(Qⁿ₊(2λ)×(0,2T)) ≤B^∗ uniformly for all small T, it is sufficient to prove

k∇xfk

W¹⁻

1p ,1 2 (1−1

p)

p (Qⁿ⁻¹(λ)×(0,T))

≤c^∗· kfk_W2,1

p (Qⁿ₊(2λ)×(0,2T))

for allf ∈C²(Qⁿ₊(2λ)×[ 0,2T]). The most difficult part in this inequality is the estimate for the time-regularity of the trace, i.e.

(2.1) |∇xf|

L^0,

12 (1−1 p)

p (Qⁿ⁻¹(λ)×(0,T))

≤c^∗· kfk_W2,1

p (Qⁿ₊(2λ)×(0,2T)),

where|g|^p

L^0,βp (Qⁿ⁻¹(λ)×( 0,T)):=

T

R

0

h^−(1+pβ)k∆_t,hgk^p_p,Qn−1(λ)×( 0,T−h)dh

forβ∈( 0,1 ), when (∆_t,hg)(x^′, t) :=g(x^′, t+h)−g(x^′, t) andk · k_{p, X} :=k · k_Lp(X). The estimate for the spatial regularity follows from the more elementary trace the- oremW_p¹(Ω)→W¹⁻

1

p p(∂Ω) (cp. KUFNER et al. [K/J/F, 6.8.13 Theorem, p. 337]) by an easy scaling argument (in t). In the sequel, we shall prove (2.1). For this purpose, we start from the representation (1.1) for ∂_jf (1 ≤ j ≤ n): splitting RT

0 (· · ·)dv =Rh

0 (· · ·)dv+RT

h (· · ·)dv in the sum in the second line in (1.1) we get

∂_jf(·) =H₁(·) +

n+1

X

i=1

B˜_i{H₂⁽ⁱ⁾(·) +H₃⁽ⁱ⁾(·)}, where

(2.2.)

H1(·) := A T^r

Z . . .

Z

Qⁿ⁺¹(0, T^κ)

f(·+y)Π(y, T)dy,

H₂⁽ⁱ⁾(·) :=

Z h

0 v^−(1+r) Z

. . . Z

Qⁿ⁺¹(0, v^κ)

∂_i^lif(·+y)·K_i(y, v)dy dv,

H₃⁽ⁱ⁾(·) :=

Z T h

v^−(1+r) Z

. . . Z

Qⁿ⁺¹(0, v^κ)

∂_i^lif(·+y)·K_i(y, v)dy dv.

In the sequel, we set (γH₁)(x^′, t) :=H₁(x^′,0, t) ; we find

(2.3) k∆_t,h(γH1)k_p,Qn−1(λ)×(0,T−h) ≤ h· k∂t(γH1)k_p,Qn−1(λ)×(0,T)

(use|∆_t,hf(τ)| ≤Rh

0 |f^′(τ+s)|dsand Minkowski’s integral inequality (cp. WHEE- DEN/ ZYGMUND [W/Z, p. 143])); now

|∂_t(γH₁)(x^′, t)| ≤ A

T^r · kΠ(·, T)k_∞,Qn+1(0, T^κ)· |Qⁿ⁺¹(0, T^κ)|^1/p′

· k∂_tf((x^′,0, t) +·)k_p,Qn+1(0, T^κ)

by (2.2) and H¨older’s inequality; hence

(2.4) ≤c^∗·T−|κ|·(1−1/p^′)−κj· k∂tf((x^′,0, t) +·)k_p,Qn+1(0, T^κ)

by the kernel-estimate (1.2). Now observe that k∂_tf((x^′,0, t) +·)k^p_{p,Qn+1(0, T}κ)=

= Z _T^κn

0

k∂_tf(x^′+·, y_n, t+·)k^p

p,Qˇⁿ⁺¹(0, T^κ)dy_n, which easily implies via Fubini’s theorem

(2.5)





 Z

. . . Z

Qⁿ⁻¹(λ)×( 0,T)

k∂tf((x^′,0, t) +·)k^p_p,Qn+1(0, T^κ)dx^′dt







1/p

≤

≤ |Qˇⁿ⁺¹(0, T^κ)|^1/pk∂tfk_p,Qn((−λ1^′,0),(λ1^′+T^κ′,T^κn))×(0,2T). Hence, by the last inequality, (2.4) and since|Qˇⁿ⁺¹(0, T^κ)|=T^|κ|−12 andκ_j = ¹₂:

r.h. side in (2.3)

≤c^∗·h·T^−12(1+p1)· k∂_tfk

p, Qⁿ((−λ1^′,0),(λ1^′+T^κ′,T^1/2))×(0,2T)

so that, abbreviatingρ=ρ(p) := ¹₂(1−_p¹),

|γH₁|

L^0,ρp (Qⁿ⁻¹(λ)×(0,T))≤

≤c^∗·T^−21(1+1p) Z T

0

h^{−1+p(1−ρ)}dh

!1/p

k∂_tfk_{p, Q}n(a, b)×(0,2T)

witha:= (−λ1^′,0) and b:= (λ1^′+T^κ′, T^1/2) ; now 1−ρ= ¹₂(1 +¹_p) and theT factors in the last inequality cancelled, as desired.

Let us turn our attention toH₂⁽ⁱ⁾: trivially, forh≤T,

(2.6) k∆_t,h(γH₂⁽ⁱ⁾)k_p,Qn−1(λ)×(0,T−h) ≤ 2· kγH₂⁽ⁱ⁾k_p,Qn−1(λ)×(0,T); furthermore, using the kernel estimate (1.3) (withs= 0), we get

(2.7) |γH₂⁽ⁱ⁾(x^′, t)| ≤

≤c^∗· Z h

0

v^{−(1+|κ|+εκn)+12} Z

. . . Z

Qⁿ⁺¹(0, v^κ)

y_n^ε· |∂_i^lif((x^′,0, t) +y)|dy dv;

we now represent the integrand as

{v^{−p1′(1+|κ|)+12(ρ−ε·κn)}} · {v^{−1p(1+|κ|−21)+12(ρ−εκn)}·y^ε_n· |∂_i^lif((x^′0, t) +y)| } (note that 1/2 =ρ+ 1/2p) ; we chooseε∈(0, ρ/κ_n); H¨older’s inequality (withp^′, p) iny-v space then yields

(2.8) l.h.s. in (2.7)≤c^∗· Z h

0 v^−1+p

′

2(ρ−ε·κn)dv

!1/p^′

·I^1/p

with I:=

Z _h

0

Z . . .

Z

Qⁿ⁺¹(0, v^κ)

v^{−(1+|κ|−12)+p2(ρ−ε·κn)}·y_n^εp· |∂_i^lif((x^′,0, t) +y)|^pdy dv ,

where in the first integral we took into account that|Qⁿ⁺¹(0, v^κ)|=v^|κ|; the first integral is clearly proportional toh^{12(ρ−ε·κn)}. Thus, after a computation as in (2.5), we get

(2.9) kγH₂⁽ⁱ⁾k_p,Qn−1(λ)×(0,T)≤c^∗·h^{12(ρ−ε·κn)}·I˜^1/p with

I˜:=

Z _h

0

v^{−(1+|κ|−12)+p2(ρ−ε·κn)}|Qˇⁿ⁺¹(0, v^κ)|

Z . . .

Z

Qⁿ⁺¹(a, b(v))

z_n^ε·p· |∂_i^lif(z)|^pdz dv ,

where a:= (−λ1^′,0,0), b(v) := (λ1^′+v^κ′, v^κn, T +v); since b(v)≤b(h), we can continue

I˜≤ Z _h

0

v^{−1+p2(ρ−ε·κn)}dv Z

. . . Z

Qⁿ⁺¹(a, b(h))

z_n^ε·p· |∂_i^lif(z)|^pdz

≤ c^∗·h^{(ρ−ε·κn)·p/2} Z _h^κn

0

z_n^ε·p·ϕ(z_n)dz_n withϕ(zn) :=k∂_i^lif(·, zn,·)k^p

p, Qⁿ⁻¹(−λ1^′, λ1^′+T^κ′)×(0,2T) by Fubini’s theorem and sinceh≤T; consequently, by (2.6), (2.9) and the last line

(2.10) |γH₂|^p

L^0,ρ_p (Qⁿ⁻¹(λ)×( 0,T))≤

≤c^∗· Z T

0

h^{−(1+p·ε·κn)} Z h^κn

0

z^ε·p_n ·ϕ(zn)dzndh

and by the version of Hardy’s inequality from Lemma, ( i) in the Appendix

≤c^∗·(p·ε·κ_n)⁻¹· Z _T^κn

0

ϕ(z_n)dz_n

=c^∗·(p·ε·κ_n)⁻¹· k∂_i^lifk^p

p, Qⁿ((−λ1^′,0),(λ1^′+T^κ′,T^1/2))×(0,2T) , which is the desired result forH₂⁽ⁱ⁾.

Finally, let us turn to H₃⁽ⁱ⁾; we again use (2.3) and observe that the correct expression for∂t(γH₃⁽ⁱ⁾) is obtained just by replacingK_i (in the definition ofH₃⁽ⁱ⁾) by∂_n+1K_i (integrate by parts); after estimating|∂_n+1K_i| according to (1.3), we arrive at

(2.11) |∂t(γH₃⁽ⁱ⁾)(x^′, t)| ≤

≤ c^∗· Z T

h

v^{−(1+|κ|+12+ε·κn)} Z

. . . Z

Qⁿ⁺¹(0, v^κ)

y_n^ε· |∂_i^lif((x^′0, t) +y)|dy dv (cp. (2.7); here the v-exponent is smaller by one, since ∂n+1K_i entails (in (1.3)) the additional factorv⁻¹); in the last integral we write the integrand in the form

{v^−p1′(1+|κ|)−(1−ρ−δ)

} · {v^{−p1(1+|κ|−12)−(εκn+δ)}·y_n^ε· |∂_i^lif(· · ·)|}

(note that−¹₂ = _2p¹ +ρ−1), where we introducedδ∈(0,1−ρ). Now apply H¨older’s inequality (withp^′, p) iny-v space and get

r.h.s. in (2.11)≤c^∗· Z _T

h

v^{−1−p′·(1−ρ−δ)}dv

!1/p^′

·J^1/p with

J :=

Z _T

h

v^{−(1+|κ|−12)−p(ε·κn+δ)} Z

. . . Z

Qⁿ⁺¹(0, v^κ)

y_n^εp· |∂_i^lif((x^′,0, t) +y)|^pdy dv; proceeding as in the argument leading from (2.8) to (2.9), the last estimate allows us to conclude

k∂t(γH₃⁽ⁱ⁾)k_{p, Q}n−1(λ)×(0,T)≤

≤c^∗·h^{−(1−ρ−δ)}· Z T

h

v^{−1−p(ε·κn+δ)} Z v^κn

0

z_n^ε·p·ϕ(zn)dzndv

!1/p

withϕ(·) as before (sincev≤T); by (2.3)

|γH₃⁽ⁱ⁾|^p

L^0,ρp (Qⁿ⁻¹(λ)×(0,T))

≤c^∗· Z _T

0

h^−1+pδ Z _T

h

v^{−1−p·(ε·κn+δ)} Z _v^κn

0

z_n^ε·p·ϕ(zn)dzndv dh

≤c^∗·(p·δ)⁻¹· Z _T

0

v^{−1−p·ε·κn} Z _v^κn

0

z^εp_n ·ϕ(z_n)dz_ndv

by Appendix, Lemma (ii); now we may continue as after (2.10) and the desired result forH₃⁽ⁱ⁾ follows.

Thus (2.1) is proved for allT ≤T0(λ) =λ².

Appendix.

We note a version of Hardy’s inequality.

Lemma. Suppose thatf ∈L1(0, T^γ)is nonnegative, 0< T ≤ ∞;ε, γ >0. Then (i) R_T

0 x^{−1−ε·γ} R_x^γ

0 y^ε·f(y)dy dx ≤ (γ·ε)⁻¹ R_T^γ

0 f(y)dy, (ii) R_T

0 x^−1+ε·γ R_T^γ

x^γ y^−ε·f(y)dy dx ≤ (γ·ε)⁻¹ R_T^γ

0 f(y)dy.

Proof: These inequalities are proved in BESOV/ IL’IN/ NIKOL’SKII [B/I/N, 2.15, p. 28] (even in a more general form) forT =∞. ForT finite they follow easily by applying the version forT =∞to the extension by zero off to IR⁺.

References

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[B/I/N]Besov O.V., Il’in V.P., Nikol’skii S.M.,Integral Representations of Functions and Imbedding Theorems, Vol. I., Wiley, 1978.

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[K/J/F]Kufner A., John O., Fuˇcik S.,Function Spaces, Leyden, Noordhoff Int. Publ. 1977.

[L/S/U]Ladyshenskaya O.A., Solonnikov V.A., Uralceva N.N,Linear and Quasilinear Equations of Parabolic Type, Am. Math. Soc., Providence, R.I. 1968.

[R] R´akosn´ık J.,Some remarks to anisotropic Sobolev spaces, I. Beitr¨age zur Analysis13(1979), 55-68.

[W] Weidemaier P.,Local existence for parabolic problems with fully nonlinear boundary con- dition; anL^p-approach, to appear in Ann. mat. pura appl..

[W/Z]Wheeden R. L., Zygmund A.,Measure and Integral., New York - Basel: Dekker 1977.

University of Bayreuth, Faculty of Mathematics and Physics, P.O.Box 101251, 8580 Bayreuth, Federal Republic of Germany

(Received August 6, 1990)